National problem · Geometry

In a hexagon \(ABCDEF\) the following hold: \(\angle BAF = 150^{\circ}\), \(\angle ACB = \angle ADC = 90^{\circ}\), \(|AC| = |BC|\), triangle \(ABC\) is similar to triangle \(ADE\), and triangle \(BCD\) is similar to triangle \(DEF\). Determine the ratio of the lengths of the segments \(AB\) and \(AF\).

The hexagon with the two diagonals from \(A\) that carry the data: right angles at \(C\) and \(D\), equal segments \(AC\) and \(BC\), and \(\angle BAF = 150^{\circ}\).

Slovenian High School Mathematics Competition for Vega Awards (MaSSA), drzavno (national) round 2011, 1. letnik, category A, problem 3. Organized by DMFA Slovenije (Society of Mathematicians, Physicists and Astronomers of Slovenia). Source

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